Acoustic Performance of a Barrier Embedded with Helmholtz

ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 103 (2017) 444 –450 DOI 10.3813/AAA.919074

Acoustic Performance of aBarrier Embedded With Helmholtz ResonatorsUsing a Quasi-Periodic BoundaryElement Technique

Samaneh M. B. Fard1),Herwig Peters1),Steﬀen Marburg2),Nicole Kessissoglou1) 1) School of Mechanical and Manufacturing Engineering, The University of NewSouth Wales, Sydney, NSW 2052, Australia. [email protected] 2) Institute of Vibroacoustics of Vehicles and Machines, Faculty of Mechanical Engineering, Tecnical University of Munich, Boltzmann Str.15, 85748 Garching, München, Germany

Summary Barriers are widely used to attenuate environmental noise from vehicles to residential areas. This paper numerically explores the acoustic performance of abarrier for tailored lowfrequencynoise reduction using aquasi- periodic boundary element method, whereby the barrier is represented by afinite number of periodic sections along its length. Results for the insertion loss of abarrier with single and multiple Helmholtz resonators embedded along the top edge of each periodic barrier section and tuned to different frequencies are compared to results for an equivalent straight barrier in the absence of the Helmholtz resonators. The acoustic performance of the noise barrier with and without the embedded Helmholtz resonators is also examined for elevated trafficnoise sources typically corresponding to engine and exhaust noise of light and heavy vehicles. Diffraction overthe top edge of the barrier and reflections from the ground on both the source and receiversides of the barrier are taken into account. ©2017 The Author(s).Published by S. Hirzel Verlag · EAA. This is an open access article under the terms of the Creative Commons Attribution (CCBY4.0)license (https://creativecommons.org/licenses/by/4.0/). PACS no. 43.50.Gf

1. Introduction random edge profile barrier does not perform as well as a conventional noise barrier at lowfrequencies. Other stud- Barriers are commonly used as acontrol measure to re- ies investigated the acoustic performance of curved noise duce outdoor sound propagation from road trafficnoise barriers [10]. Incorporating cavities or resonators in barrier affecting nearby residential areas. The amount of reduc- designs is another approach to attenuate noise at particular tion largely depends on the barrier geometry and material frequencies [7, 11, 12, 13]. High levels of attenuation were properties. The simplest method to increase the efficiency presented for awave-trapping barrier by Pan et al. [7]. The of anoise barrier is generally to increase its height, but wave-trapping barrier comprised of multiple wedges with such increases are not always practical due to aesthetics, perforated surfaces, aback cavity and an internal lining, cost, maintenance and safety reasons. Forexample, barri- and wasshown to significantly reduce the sound pressure ers subject to wind loads have height constraints, thereby levelatlow frequencies. reducing their effectiveness. Theoretical methods for estimating the sound attenu- There are anumber of different designs to the top of ation produced by anoise barrier are well established, abarrier in an attempt to modify the diffracted wavesand for example, see [14, 15, 16]. The finite element method improve barrier performance compared to astraight barrier (FEM)has been used to compute the insertion loss of of similar height [1, 2, 3, 4, 5, 6, 7, 8]. Areviewofprofiled anoise barrier [17, 18]. Tyurina et al. [17] developed noise barriers including T-shaped, Y-shaped and arrow- an FEM model for a4mhigh noise barrier and vali- shaped barriers revealed that in most cases, aT-shaped bar- dated their numerical results with experimental tests. An rier performs better than the other noise barrier designs [3]. FEM acoustic topology optimisation model wasdevel- Shao et al. [9] showed that abarrier with arandom edge oped to design anoise barrier with minimum Zwickers profile along the top improvedbarrier performance com- loudness [18]. Anumerical approach that can satisfy the pared to astraight barrier.However,itwas shown that the Sommerfeld radiation condition is the boundary element method (BEM), which has been widely used to predict the acoustic performance of arange of barrier designs Received2November 2015, [1, 2, 10, 19, 20, 21, 22]. Boundary integral equations were accepted 23 March 2017. solved using the BEM and reasonable agreement between

©2017 The Author(s). Published by S. Hirzel Verlag · EAA. 444 This is an open access article under the terms of the CC BY 4.0 license. Fard et al.:Barrier embedded with Helmholtz resonators ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 103 (2017) predicted and measured insertion loss values wasobtained [1, 2, 19, 20]. Lam [21] modified the BEM formulation to predict the insertion loss of abarrier incorporating attenuation due to atmospheric turbulence. Gasparoni et al. [22] investigated fork shaped barriers using an openBEM Figure 1. Illustration of the quasi-periodic boundary. software. BEM simulations of barriers were implemented into noise mapping software whereby the acoustic performance of several complexbarrier designs were compared with astraight barrier of equal height [10]. To save computational cost, periodic and quasi-periodic structures have been used to solvemodels of three-dimensional (3D) infinite structures. Hashimoto et al. [23] tested an optimization analysis for periodic sound barriers composed of elastic scatterers and subject to plane wave exci- tation. Lam [24] numerically studied plane wave incidence on the surface of aperiodic structure. 3D plane wave scattering in doubly periodic structures wasinvestigated using Floquet theory [25]. In recent work, Ziegelwanger et al. [26] coupled the fast multiple method with aquasi- periodic boundary element method to simulate the sound field scattered around anoise barrier.Karimi et al. [27] compared the insertion loss for asonic crystal barrier using Figure 2. Configuration of asingle periodic barrier section show- aperiodic BEM technique, whereby aunit cell comprised ing monopole sources. arow of cylindrical scatterers. In this work, a3Dquasi-periodic BEM technique re- cently developed by the authors [28] is implemented to The acoustic pressure is assumed to be periodic, such investigate the acoustic performance of abarrier with em- that the acoustic pressure at each collocation point on a bedded Helmholtz resonators tuned to asingle frequency typical boundary section is equal to the acoustic pressure or multiple frequencies. Only asmall section of the bar- at the corresponding point on the other sections. Hence, rier is considered and periodically repeated in one direc- only the acoustic pressures at the collocation points of the tion, thus significantly reducing the simulation run times centre boundary Γ0 need to be determined. and storage requirements compared to afull 3D model. The barrier wasmodelled using four-node super-para- Diffraction overthe top edge of the barrier and reflections metric discontinuous linear boundary elements. Aconver- from the ground on both the source and receiversides of gence analysis to identify the required number of elements the barrier are taken into account. Results are compared per wavelength wasconducted. Atruncation analysis was to those for an equivalent straight barrier in the absence also carried out to identify the minimum number of bar- of the Helmholtz resonators and for elevated trafficnoise rier sections for the quasi-periodic model to represent an sources. infinitely long barrier.The full description of the quasi- periodic BEM technique is givenin[28].

2. Numerical methods 2.2. Finite element model 2.1. Quasi-periodic boundary element method Three-dimensional barrier models were also developed using commercial finite element software COMSOL Multi- In the quasi-periodic boundary element method, the boun- physics (v4.3b)tovalidate the convergence analysis re- dary of the barrier Γ is divided into small sections Γ as n sults for the quasi-periodic BEM. Comparisons were only shown in Figure 1. The boundary Γ is then represented by presented for abarrier with single Helmholtz resonator due summation of the boundary sections Γ as n to limitations in computational resources for the finite element model. Aperfectly matched layer (PML)was used Γ=Γ−M∪Γ−M+1∪···∪Γ−1 ∪Γ0 ∪Γ1 ∪... to approximate radiation into the infinite exterior acous- ···∪Γ ∪···∪Γ ∪Γ . (1) n N−1 N tic domain. The thickness of the PML wasscaled with N and M are the number of boundaries on each side of the frequencysuch that its thickness wasequal to one wavelength. The acoustic domain in the FE model wasdis- initial boundary Γ0.From the boundary condition givenby equation (1),the total length of the barrier is givenby cretized using quadratic Lagrangian elements. To achieve an accurate prediction of pressure overafinite element,

LT =(N+M+1)xp. (2) the dimensions of the elements were chosen to be less than one ﬁfth of an acoustic wavelength. Aperiodic boundary where xp is the length of one section of the periodic barrier, condition wasapplied at the lateral edges, thus extending as shown in Figure 2. the length of the barrier to inﬁnity in one dimension.

445 ACTA ACUSTICA UNITED WITH ACUSTICA Fard et al.:Barrier embedded with Helmholtz resonators Vol. 103 (2017)

3. Barrier models

Aperiodic section of arectangular barrier with athick- ness of 0.5 mand aheight of 3m wasinitially modelled. The length of each periodic section was1m. Asingle Helmholtz resonator wasthen embedded along the top part of each barrier section. The cross section of the opening port wasasquare and the cavity wasarectangular shape. The Helmholtz resonator wastuned to aspecificfrequency by adjusting the size of the resonator cavity and opening and length of the neck. Figure 3shows the configuration of asingle periodic barrier section with and without aHelmholtz resonator,modelled using the quasi-periodic BEM technique. Twodifferent Helmholtz resonator geometries were considered where each geometry wastuned to the same Figure 3. Aperiodic barrier section without (left)and with (right) frequency. One geometry corresponded to aHelmholtz aHelmholtz resonator. resonator with alarge cavity and alarge neck opening butashort neck. The second geometry corresponded to 6 aHelmholtz resonator with asmaller cavity and asmaller 10 neck opening butalonger neck. Forboth geometries, the 20

(Pa) 35 ratio of the side length of the neck opening to the length of 4 the cavity wasconstant.

The Helmholtz resonator frequency fHR is givenby[29, pressure 2 30] Sound 0 c A 100 150 200 250 300 s Frequency (Hz) fHR = , (3) 2π ls,effVcav Figure 4. Sound pressure as afunction of frequencyfor different where c is the speed of sound in air, As is the area of the number of elements. neck opening, Vcav is the volume of the resonator cavity and l is the effective neck length of the Helmholtz res- s,eff and as such wasselected for all subsequent barrier models onator.The effective length of the neck ls,eff is used to account for the fact that some extra volume of air around the with embedded Helmholtz resonators. neck moveswith the air inside the neck and is givenby In both the FEM and quasi-periodic BEM models, ten [29, 30] monopole sources in close proximity were associated with each periodic barrier section. The monopole sources were l located at aheight of 0.01 mabove the ground and at a l = l + 1.7√0 , (4) s,eff s π normal distance of 1m from the mid-plane of each periodic section, as shown in Figure 2. The receiverislocated where ls is the length of the neck and l0 is aside length on the ground 5mfrom the mid-plane of the barrier in the of the square neck opening. In the quasi-periodic bound- shadowzone. Results are presented in terms of barrier in- ary element model, an equal number of periodic sections sertion loss corresponding to M = N = 400 were used on each side pg of the centre section, resulting in atotal barrier length of IL = 20 log , (5) 801 m. Aconvergence analysis wascarried out to identify pb the required number of elements to model the periodic bar- where pg is the acoustic pressure radiated from asource in rier section with an embedded Helmholtz resonator using the presence of aground surface only (inthe absence of the quasi-periodic BEM. The barrier wasmeshed such that the barrier)and pb is the acoustic pressure with the barrier the boundary elements on the side edges of the barrier had in place at the same receiverlocation. the same size. The top edge of the barrier and all the edges of the Helmholtz resonators have equal or higher mesh density than the barrier side edges. Three different mesh 4. Results and discussion sizes were considered, corresponding to approximately 10, 4.1. Helmholtz resonators tuned to asingle fre- 20 and 35 elements per wavelength. As shown in Figure 4, quency results for the acoustic pressure as afunction of frequency at areceiverlocated on the ground 5mfrom the mid-plane Results for the insertion loss for abarrier with one embed- of the barrier in the shadowzone wasfound to have con- ded Helmholtz resonator per periodic section developed verged for the mesh size of 20 elements per wavelength, using the quasi-periodic BEM technique are compared to

446 Fard et al.:Barrier embedded with Helmholtz resonators ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 103 (2017) results obtained for asimilar barrier developed using the FEM. Figure 5presents the insertion loss for aquasi- 25 periodic barrier in the absence of Helmholtz resonators (a) 20 and the barrier with asingle Helmholtz resonator embed-

(dB) ded in the top surface of each periodic section. Results for 15 barriers with embedded Helmholtz resonators tuned to fre- loss 10 No HR quencies of 200 Hz or 250 Hz are presented. Results are BEM (200 Hz) 5 also presented for the twodiﬀerent Helmholtz resonator Insertion BEM (250 Hz) geometries corresponding to aresonator with alarger cav- 0 FEM (200 Hz) 100 150 200 250 300 ity and short neck or asmaller cavity and long neck. Fora Frequency (Hz)

Helmholtz resonator with the small cavity/long neck, the 25 insertion loss peaks occur exactly at the tuned frequency (b) in anarrowfrequencyrange (Figure 5a). Forthe large 20

(dB) cavity/short neck, there is aslight shift in the peak inser- 15 tion loss from the tuned frequencytoahigher frequency loss 10 No HR (Figure 5b). The increase in insertion loss compared to a BEM (200Hz) 5 barrier without Helmholtz resonators also occurs overa Insertion BEM (250Hz) FEM (200 Hz) broader frequencyrange. 0 100 150 200 250 300 Results obtained from the 3D FE model are also pre- Frequency (Hz) sented for the Helmholtz resonator tuned to aresonant frequencyof200 Hz for the twodiﬀerent resonator ge- Figure 5. Insertion loss of aquasi-periodic barrier without and ometries. Very close agreement in the results obtained us- with aHelmholtz resonator tuned to aresonant frequencyof ing the quasi-periodic BEM and FEM is observed in Fig- 200 Hz or 250 Hz with (a) asmall cavity/long neck and (b) a ure 5b for the Helmholtz resonator with alarge cavity and large cavity/short neck. short neck. Forthe Helmholtz resonator with asmall cavity and longer neck (Figure 5a), aslight shift in frequency at which peak insertion loss occurs can be observed, attributed to the size of boundary elements. When the number of elements around the Helmholtz resonator in the quasi-periodic BEM model are increased, the results con- vergetothose obtained from FEM. The acoustic performance of the barrier modelled using the quasi-periodic BEM technique, with and without aHelmholtz resonator embedded in each periodic section, for arange of receiverpositions in the barrier shadowzone is presented in Figure 6. The shadowzone extends to anormal distance of 10 mfrom the barrier (inthe y-direction). The Helmholtz resonator geometry of alarge cavity/short neck tuned to afrequencyof200 Hz wasused. In Fig- ure 6, the insertion loss results are at 210 Hz, which is the frequencyatwhich the peak insertion loss in Figure 5b occurs. With and without the presence of Helmholtz resonators, there is agradual reduction in insertion loss with increasing normal distance from the barrier.For the barrier in the absence of Helmholtz resonators (Figure 6a), at least 12 dB attenuation can be observed in the entire shadowzone. Forthe barrier with embedded Helmholtz Figure 6. Insertion loss at 210 Hz in the shadowzone of aquasi- resonators, signiﬁcantly greater insertion loss of at least periodic barrier (a) without and (b) with embedded Helmholtz 20 dB in the barrier shadowzone is achieved(Figure 6b). resonators tuned to afrequencyof200 Hz.

4.2. Helmholtz resonators tuned to multiple fre- Forone set of results, the small cavity/long neck geom- quencies etry wastuned to 200 Hz whilst the large cavity/short neck Figure 7presents insertion loss results without embed- geometry wastuned to 250 Hz. In another set of results, ded Helmholtz resonators and with twoHelmholtz res- only the large cavity/short neck geometry wasused, one onators per periodic section tuned to diﬀerent frequencies, tuned to 200 Hz and the other to 250 Hz. The use of two whereby one resonator is tuned to afrequencyof200 Hz Helmholtz resonators per periodic section tuned to diﬀer- and the other is tuned to 250 Hz. Furthermore, the twodif- ent frequencies results in twopeaks in the insertion loss. ferent Helmholtz resonator geometries were considered. Similar to the results in Figure 5, for the large cavity/short

447 ACTA ACUSTICA UNITED WITH ACUSTICA Fard et al.:Barrier embedded with Helmholtz resonators Vol. 103 (2017)

25 25

20 20

(dB)

(dB) 15 15

loss

loss 10 10 No HR No HR 3HR (150, 172 and 250 Hz)

Insertion 5 2HR (long neck 200 Hz, short neck 250 Hz) Insertion 5 3HR(150, 162 and 250 Hz) 2HR (short neck 200 Hz, short neck 250 Hz) 3HR(150, 156 and 250 Hz) 0 0 100 150 200 250 300 100 150 200 250 300 Frequency (Hz) Frequency (Hz)

Figure 7. Insertion loss of aquasi-periodic barrier without and Figure 8. Insertion loss of aquasi-periodic barrier without and with twoHelmholtz resonators tuned to frequencies of 200 Hz with multiple Helmholtz resonators tuned to frequencies of and 250 Hz. 150 Hz, 200 Hz and 250 Hz. neck geometry,there is abroader increase in insertion loss and ashift in peak frequencyatwhich peak insertion loss 25 occurs, compared with the small cavity/long neck geometry.Figure 7shows that the use of twoHelmholtz res- 20

(dB) onators in close proximity and tuned to diﬀerent frequen- 15 cies can result in aglobal increase in insertion loss be- loss 10 tween the selected tuned frequencies compared to the in- 5 sertion loss for abarrier in the absence of Helmholtz res- Insertion No HR 3HR (150, 200 and 250 Hz) onators. In Figure 8, the insertion loss for aquasi-periodic 0 100 150 200 250 300 barrier with three Helmholtz resonators per periodic sec- Frequency (Hz) tion is presented, whereby resonators tuned to frequencies of 150 Hz, 200 Hz and 250 Hz are embedded in series Figure 9. Insertion loss of aquasi-periodic barrier without and in each periodic barrier section. All Helmholtz resonators with multiple Helmholtz resonators tuned to diﬀerent frequen- have the same large cavity/short neck geometry.Introduc- cies. ing the third Helmholtz resonator tuned to alower frequencyresults in alarge increase in insertion loss around the selected frequency, although alarge dip before the next 25 tuned frequencyalso occurs. In an attempt to remove the dip, the tuned frequencyofthe Helmholtz resonator asso- 20

(dB) ciated with the middle frequencyisshifted to lower and 15 higher frequencies. Figure 9shows that tuning the middle loss 10 No HR Helmholtz resonator to alower frequency, the dip in the 3HR (150, 200 and 250 Hz)

insertion loss disappears and broad insertion loss overthe Insertion 5 3HR(150, 225 and 250 Hz) 3HR(150, 235 and 250 Hz) entire frequencyrange between the lower and upper tuned 0 100 150 200 250 300 frequencies of the Helmholtz resonators is achieved. How- Frequency (Hz) ever,tuning the Helmholtz resonator associated with the middle frequencytoahigher frequencyresults in ashift Figure 10. Insertion loss of aquasi-periodic barrier without and in the dip to higher frequencies (Figure 10). Figure 11 with multiple Helmholtz resonators tuned to different frequen- compares the insertion loss for twoorthree Helmholtz cies. resonators per periodic section, where the lower and upper frequencies of the embedded Helmholtz resonators are the same. Greater broadband insertion loss using three and 0.5 m, 1.5 mand 3.5 mabove the ground. The receiver Helmholtz resonators per periodic section can be achieved is located on the ground plane at ahorizontal distance of when the middle frequencyistuned just above the lower 10 mfrom the barrier.When both the source and the re- frequency. ceiverare on the ground, there are no interference effects from the ground and the insertion loss steadily increases 4.3. Effect of an elevated source with increasing frequency. Peaks in insertion loss for an elevated source and the receiveronthe ground, attributed The effect of an elevated source on the barrier acoustic to destructive interference effects between direct and re- performance is initially examined using a2Dfinite ele- flected wavesonthe source side of the barrier,occur at ment model. The effect of varying the height of the source discrete frequencies givenby for the 3m high straight barrier with astem thickness of

0.1 mispresented in Figure 12. The source is located at a c(ns+0.5) horizontal distance of 5mfrom the barrier,onthe ground fs= ,ns=0,1,2,..., (6) ds + dr − rs

448 Fard et al.:Barrier embedded with Helmholtz resonators ACTA ACUSTICA UNITED WITH ACUSTICA Vol. 103 (2017)

25 45 40 On ground 0.5 m 1.5 m 2.5 m 20 35

(dB) (dB) 30 15 25

loss

loss 20 10 No HR 15 5 10

Insertion Insertion 2HR (150 and 250 Hz) 3HR (150, 156 and 250 Hz) 5 0 0 100 150 200 250 300 0 250 500 750 1000 1250 1500 Frequency (Hz) Frequency (Hz)

Figure 11. Insertion loss of aquasi-periodic barrier without and Figure 12. Insertion loss of a3mstraight barrier for diﬀerent with multiple Helmholtz resonators tuned to diﬀerent frequen- source vertical locations. cies.

45 where rs is the distance from the source to the top of the 40 On ground (no HR) On ground 0.25 m 0.5 m 0.75 m 1m barrier, ds is the distance from the source to the ground 35 plane, and d is the distance the reﬂected wave travels from (dB) 30 r 25

loss the ground plane to the top of the barrier.Atfrequencies 20 givenbyequation (6),the direct and reflected waveson 15 10 the source side of the barrier have aphase difference close Insertion 5 to π. 0 50 100 150 200 250 300 Forthe source elevated 0.5 moffthe ground, large peaks Frequency (Hz) in insertion loss equispaced on the frequencyscale occur at frequencies givenbyEquation (6),with the minimum Figure 13. Insertion loss of a3mstraight barrier with and without values for insertion loss between successive peak frequen- embedded Helmholtz resonators tuned to afrequencyof200 Hz cies corresponding to the insertion loss for asource on for different source vertical locations obtained using the quasi- the ground. As the source height increases, the number of periodic boundary element method. equispaced peaks in insertion loss increases with acorre- sponding decrease in insertion loss as well as adecrease in frequencyatwhich the first peak occurs. When the source When the peak in insertion loss due to the elevated sources height is approaching the height of the barrier,the inser- occurs at higher frequencies than the tuned Helmholtz res- tion loss is gradually reduced to amid-range around the onator frequency, aglobal increase in insertion loss be- insertion loss for the source on the ground. tween the twopeak frequencies can be observed. However, when the peak in insertion loss due to the elevated sources The acoustic performance of abarrier with asingle occurs at lower frequencies than the tuned Helmholtz res- Helmholtz resonator embedded in each periodic section, onator frequency, the presence of the Helmholtz resonance modelled using the 3D quasi-periodic BEM technique, is does not appear to significantly affect the peak due to the nowexamined for sources on and above the ground. Re- elevated sources except when the twofrequencies are in sults for the insertion loss for abarrier with astem thick- close proximity.Due to the limited frequencyrange in Fig- ness of 0.5 m, with and without an embedded Helmholtz ure 13, multiple peaks in insertion loss due to elevated resonator are shown in Figure 13. Tenmonopole sources sources only appear for the source 1moffthe ground. in close proximity were associated with each periodic barrier section, where the monopole sources were located at heights of 0.01 m(corresponding to nominally on the 5. Conclusions ground), 0.25 m, 0.5 m, 0.75 mand 1mabove the ground and at anormal distance of 1mfrom the mid-plane of each Numerical models to assess the acoustic performance periodic section. The receiverislocated on the ground 5m of athree-dimensional noise barrier for sources on the from the mid-plane of the barrier in the shadowzone. The ground were explored using aquasi-periodic BEM tech- Helmholtz resonator is tuned to afrequencyof200 Hz. nique. One Helmholtz resonator tuned to aspecificfre- In Figure 13, the presence of the embedded Helmholtz quencywas incorporated into each periodic section of the resonators results in apeak in insertion loss just above the barrier.Greater insertion loss wasachievedfor the bar- tuned frequency. Asecond peak in insertion loss occurs for rier with an embedded Helmholtz resonator around the the elevated sources. Forsources 0.25 mabove the ground, tuned frequencycompared to the barrier without embed- the second peak due to the elevated sources occurs be- ded Helmholtz resonators. The use of multiple Helmholtz yond the frequencyscale shown in the figure. The peak resonators in each periodic section tuned to different fre- in insertion loss due to the elevated sources decreases with quencies wasalso investigated, achieving greater perfor- frequencywith an increase in height of sources above the mance overabroadband frequencyrange. The acous- ground. From Figure 13, twoobservations are as follows. tic performance of the barrier with embedded Helmholtz

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